¡Descarga Economic Equations: Unemployment, Inflation, and Capital Growth - Prof. Pascali y más Apuntes en PDF de Administración de Empresas solo en Docsity! SOLUTIONS PROBLEM SET n. 8 1. Consider an economy characterized by the following relations: ut – ut-1 = – 0.4 (gyt – gy*) [Okun’s law] πt = πte + 0.05 – ut [Phillips curve] gyt = gm – πt [aggregate demand] where gyt and gm represent growth rates of output and nominal money, respectively, and g y* is the normal growth rate of output, which in this exercise we assume to be zero (gy* = 0). a) Suppose that initially the economy is in a medium run equilibrium with a growth rate of money gm = 5%. What are the values of the unemployment rate, the growth rate of output, and the inflation rate? In a medium run equilibrium πt = πte , ut = ut-1 = un , gyt = gy*. So from the Phillips curve: ut = 0.05 = 5% ; from Okun’s law: gyt = 0 Substituting in the aggregate demand: πt = 0.05 = 5% Consider a permanent increase (starting in period t) in the growth rate of money, which becomes now gm = 10%. b) Short run effects (in period t, assuming expectations have not changed, that is π te = πt-1): What are the values of the unemployment rate, the growth rate of output, and the inflation rate in period t ? [Hint: substitute the growth rate of output from the aggregate demand into Okun’s law; then substitute the inflation rate from the Phillips curve; etc.] A more expansionary monetary policy increases in the short run the growth rate of output (through aggregate demand), causing a reduction in the unemployment rate (Okun’s law), which in turn causes an increase in inflation (Phillips curve), which limits the expansionary effect on gyt. Substituting aggregate demand and Phillips curve into Okun’s law: ut – ut-1 = – 0.4 (gm – πt – gy*) = – 0.4 (gm – (πte + 0.05 – ut)) In the short run agents have not adjusted their expectations, so π te = πt-1 = 0.05 → ut – 0.05 = – 0.4 (0.10 – (0.05 + 0.05 – ut)) → 1.4ut = 0.05 → ut = 0.036 = 3.6% Substituting into the Phillips relation: πt = πte + 0.05 – ut = 0.05 + 0.05 – 0.036 → πt = 6.4% In the aggregate demand: gyt = 0.10 – 0.064 → gyt = 3.6% c) Medium run effects (when expectations are fulfilled, and wages have fully adjusted, that is with πte = π t): What are the values of the unemployment rate, the growth rate of output, and the inflation rate in the new medium run equilibrium? In the medium run, again πt = πte and ut = ut-1. From Phillips curve: ut = 5% ; from Okun’s law: gyt = 0 ; from aggregate demand, with gm =10%: πt = 10% --------- 2. Assume that: the aggregate production function takes the form with 0<<1; capital depreciates at a rate ; there is no government spending or taxes; no technological change; the labor force is constant, i.e. N t = N; aggregate saving is a constant fraction s (the saving rate) of aggregate output. Denote with k t the amount of capital per worker (Kt /N) and with yt output per worker (Yt /N). a) What is the marginal product of capital (Yt /Kt)? Is it constant or decreasing in Kt? Write the production function in per capita (per worker) terms, i.e. yt as a function of kt. --- decreasing in Kt --- b) Obtain the equation that determines the change in capital per worker (kt+1 – kt) and, from that, the equation that determines the growth rate of capital per worker (k t+1 – 1 kt) / kt. Then show the graphical representation of each, indicating the steady state level k*. --- Divide by N and use : (the absolute change in k is the difference between the investment curve and the depreciation line) Divide by k : (the growth rate of k is the difference between the investment/capital ratio curve and the depreciation rate) --- c) How is the growth rate of output per worker (yt+1 – yt) / yt related to the growth rate of capital per worker (kt+1 – kt) / kt ? --- Denote with gyt the growth rate of output per worker, with gkt the growth rate of capital per worker, so that yt+1 = (1+ gyt) yt and kt+1 = (1+ gkt) kt : , --- d) Solve for the steady state level of capital per worker k* and output per worker y*, as a function of the parameters of the model (, , s). Then compute the value of k* and y* if, for example, = 1/2, = 1%, and s = 0.10. --- Steady state: kt+1 = kt. From and . For = 1/2, = 1% and s = 0.10 : k* = 100 , y* = 10 . --- e) Assume the economy is at the steady state when, in period T, a permanent increase in the saving rate s occurs (i.e. the new saving rate is s > s). Show analytically (with 2 k t*
